Question

The fox population in a certain region has an annual growth rate of 6 percent per year. It is estimated that the population in the year 2000 was 19400 . (a) Find a function that models the population t vears after 2000(t=0 for 2000). Your answer is P(t)= (b) Use the function from part (a) to estimate the fox population in the vear 2008. Your answer is (the answer should be an integer)

Answer 1

Main Answer:

(a)
`\( P(t) = 19400 * (1 + 0.06)^t \)`

(b)
`\( P(8) \approx 28804 \)`

Step-by-step explanation:

The function
`\( P(t) = 19400` times
`(1 + 0.06)^t \)` models the fox population t years after 2000. In this formula, 19400 represents the initial population in the year 2000, and the term
`\( (1 + 0.06)^t \)` accounts for the 6 percent annual growth rate. The variable 't' represents the number of years after 2000. The formula can be understood by recognizing that each year, the population is multiplied by the growth factor (1 + 0.06), reflecting the 6 percent increase.

For part (b), estimating the fox population in 2008 involves substituting
`\( t = 8 \)` into the function.
`\( P(8) \`approx 19400 times
`(1 + 0.06)^8 \)`, which yields an approximate population of 28804.

In summary, the function
`\( P(t) = 19400 \`times
`(1 + 0.06)^t \)` serves as a model for the fox population, and using it, we estimate the population in 2008 to be approximately 28804. This estimation is based on compounding growth, where the population experiences a 6 percent increase annually.